Within the well-known framework of financial portfolio optimization, we analyze the existing relationships between the condition of arbitrage and the utility maximization in presence of insider information. We assume that, since the initial time, the information flow is altered by adding the knowledge of an additional random variable including future information. In this context we study the utility maximization problem under the logarithmic and the Constant Relative Risk Aversion (CRRA) utilities, with and without the restriction of no temporary-bankruptcy. In particular, we show that the value of the insider information may be bounded while the arbitrage condition holds and we prove that the insider information does not always imply arbitrage for the insider by providing an explicit example.
Keywords: No Free Lunch Vanishing Risk; arbitrage; enlargement of filtration; optimal portfolio; risk neutral measure; value of the information.